Kinetics of HEX-BCC Transition in a Triblock Copolymer in a Selective Solvent:

Time Resolved Small Angle X-ray Scattering Measurements and Model Calculations.

Minghai Li, Yongsheng Liu, Huifen Nie, Rama Bansil*

Department of Physics, Boston University, Boston, MA 02215, USA

Milos Steinhart

Institute of Macromolecular Chemistry, Academy of Sciences of the Czech Republic, Heyrovsky Sq. 2, 162 06 Prague 6, Czech Republic

Abstract

Time-resolved small angle x-ray scattering (SAXS) was used to examine the kinetics of the

transition from HEX cylinders to BCC spheres at various temperatures in poly(styrene-b-

ethylene-co-butylene-b-styrene) (SEBS) in mineral oil, a selective solvent for the middle EB

block. Temperature-ramp SAXS and rheology measurements show the HEX to BCC order-order

transition (OOT) at ~127 oC and order-disorder transition (ODT) at ~180 oC. We also observed

the metastability limit of HEX in BCC with a spinodal temperature, Ts ~ 150 oC. The OOT

exhibits 3 stages and occurs via a nucleation and growth mechanism when the final temperature

Tf < Ts. Spinodal decomposition in a continuous ordering system was seen when Ts< Tf < TODT.

We observed that HEX cylinders transform to disordered spheres via a transient BCC state. We

develop a geometrical model of coupled anisotropic fluctuations and calculate the scattering

which shows very good agreement with the SAXS data. The splitting of the primary peak into

two peaks when the cylinder spacing and modulation wavelength are incommensurate predicted

by the model is confirmed by analysis of the SAXS data.

* Author to whom correspondence should be addressed. Email: rb@bu.edu

1

Introduction

The transition of cylindrical micelles packed in a 2-dimensional hexagonal lattice (HEX) to

spherical micelles on a BCC lattice has been studied with block copolymers quite extensively. In

particular, the HEX cylinder to BCC sphere order-order transition (OOT) has been studied in di-

and tri-block copolymer melts using small angle X-ray scattering (SAXS), small angle neutron

scattering (SANS), transmission electron microscopy (TEM), rheology, and birefringence

measurements 1, , , , ,2 3 4 5 6, , , , , 7 8 9 10 11. The HEX cylinder phase is observed at lower temperature

and BCC sphere phase emerges at a higher temperature. This transition is found to be thermally

reversible. The cylinder to cubic spheres OOT has also been studied in block copolymer

solutions12, ,13 14, 15, ,16 17, 18. The presence of a selective solvent can influence the temperature

dependence of the phase diagram; for example in poly (styrene-b-isoprene) (SI) in

diethylphthalate (DEP), a styrene-selective solvent, the cylinder to sphere OOT occurs upon

cooling , whereas in the melt it appears on heating. SANS and SAXS measurements on sheared

oriented samples of poly(ethylene propylene-b-ethyl ethylene) (PEP-PEE) diblocks , and on

single grains of SI diblock melt as well as on sheared oriented samples of SIS triblock 5, , , 6 7 8

reveal that the transition is epitaxial, with the axis of the cylinder becoming the <111> direction

of the BCC lattice, and the (100) HEX planes becoming the (110) planes of BCC. However, to

the best of our knowledge there are no measurements of the time-evolution of the HEX-BCC

transition, although the kinetics of the reverse transition, BCC to HEX, which is much slower,

has been investigated recently 19. Although the phases are thermally reversible, the mechanism

by which cylinders break up into spheres and form a BCC lattice is expected to be quite different

2

than that involved in the reverse transition because the rate of formation of cylinders by merging

spheres is much slower than the rate of breaking cylinders up.

It is generally accepted that the transition of HEX cylinders to BCC spheres involves the

formation of undulated cylinders whose radii are modulated along the cylinder axis1, , , , , , 4 5 6 7 8 16.

By allowing for anisotropic composition fluctuations Laradji et al 20 obtained modulated

cylinders in their calculations of phase diagrams of a diblock copolymer melt showing the limits

of metastability of the different ordered phases, and made predictions concerning the stability of

gyroid and hexagonally perforated lamellar phases. Recently Ranjan and Morse21 have re-

examined the instability of the gyroid phase and the possibility of transitioning from BCC

directly toward HEX instead of via a metastable perforated lamellar phase as suggested by

Laradji et al. These self consistent field calculations, as well as the time dependent Ginzburg

Landau (TDGL) calculations of the kinetics of the HEX to BCC transition also support the

occurrence of modulated cylinders22. Matsen23 used self-consistent mean field theory to examine

the pathway of the cylinder to sphere epitaxial transition and showed a nucleation and growth

mechanism with strong fluctuation effects due to the small energy barrier of the transition. He

also noted a narrow window in which spinodal mechanism would occur. Such rippled cylinders

were seen by Ryu and Lodge6, 7 using TEM and SAXS in an oriented SIS melt. Recently

Bendejacq et al 24 have obtained high-resolution TEM images of rippled cylinders in

poly(styrene-b-acrylic acid) (PS-PAA) diblocks dispersed in water, which enabled them to

measure structural parameters, such as the wavelength of the ripple (λ), the radius of the core (Rc)

and the height (h) of the brush (related to the amplitude of the fluctuation). From these

measurements they concluded that the ratio of the height of the cylindrical PAA brush to its core

3

radius determines the separation between undulating cylinders, straight cylinders and spheres.

Specifically they found that straight cylinders are found in the case h/Rc ≤ 1.8, undulating

cylinders between 1.8 < h/Rc < 2.0 and spheres above h/Rc ≥ 2.0. Their measurements clearly

support the criterion of a critical curvature as driving the transition from cylinders to spheres. A

theoretical study on the same system by Grason et al25 shows that the modulated cylinder is a

metastable state in the cylinder to sphere transition under a certain range of charge and salt

concentration where the sphere state is the thermodynamically favored stable state.

The breakup of a cylinder into spheres without any underlying lattice has been studied

extensively in the so-called “pearling instability”, according to which the amplitude of a

transverse wave along the length of the cylinder grows causing the cylinder to break up into

droplets (“pearls”). This is observed in the classical experiments of Rayleigh 26 , 27 where a

column of liquid pinches into drops, as well as in lipid vesicles in an optical laser tweezer trap28,

29 . The growth of the instability involves the competition between the surface energy and

bending elasticity of the cylinder27, , , , 28 30 31 32.

In the block copolymer case the transition involves both the breaking of cylinders to spheres

and the epitaxial transformation of the underlying lattice. While previous experimental and

theoretical studies provide insight into the epitaxial mechanism, and rippled cylinders have been

observed by TEM, there are no measurements of the time evolution of the transformation, nor

are there any reports of a formalism to calculate the azimuthally averaged scattering intensity

from rippled cylinders in an un-oriented system. Time-resolved SAXS provides a convenient

probe of the transformation kinetics. Unlike the direct visualization afforded by TEM and atomic

4

force microscopy (AFM) methods, extraction of spatial structural information from SAXS data is

not so direct. One needs a geometrical model so as to be able to correlate features in the

momentum-space scattering data with the real-space morphology of the system.

In Part I of this paper we report synchrotron based time-resolved SAXS measurements of the

kinetics of the transformation from HEX to BCC in the triblock copolymer of poly(styrene-b-

ethylene-co-butylene-b-styrene) (SEBS) in mineral oil, a selective solvent for the middle PEB

block. This system forms a network of micelles with PS in the cores and the solvated PEB chains

forming loops and bridges. Because the solvent is poor for the minority PS block it further

enhances the microphase separation tendency due to the incompatibility of PS and PEB blocks.

At a concentration of 45% the system exhibits a HEX phase at lower temperatures than the BCC

phase. This behavior is similar to that of SEBS in the melt. We examine the kinetics of HEX to

BCC transition for different values of the final temperature, as well as the HEX to disorder

spherical micelle transition. From an analysis of these data we obtain detailed insight into the

mechanism of the transition and the temperature dependence of the kinetics. In the second part of

this paper, we develop the structural model of rippled cylinders to calculate the scattering and

compare with the experimental results.

Experimental Section

Materials. A 45% w/v solution of SEBS triblock copolymer (Shell Chemicals, Kraton

G1650) with a molecular weight Mn of 100,000 Daltons, polydispersity Mw/Mn of 1.05, styrene

fraction 28 wt%, and E:B ratio 1:1 was prepared in mineral oil (J.T. Baker) which is selective to

5

the middle PEB block. Methylene chloride was used as a co-solvent to make a homogeneous

solution and then was removed by evaporation until no further change in weight was observed.

Rheology. The dynamic storage and loss moduli G΄ and G˝ were measured as a function of

temperature on an AR-G2 rheometer (TA instruments) at the Hatsopoulos Microfluids

Laboratory at MIT. We used an angular frequency ω of 1 rad/s and strain γ0 of 2% as these

parameters correspond to the linear viscoelastic regime. For the heating process a controlled

temperature ramp rate of 1 oC/min was used.

Figure 1. Temperature dependence of the dynamic shear moduli G΄ and G˝ (at ω = 1 rad/s,

and strain γ0 = 2%) from SEBS 45% in mineral oil at a heating rate of 1 oC/min.

The temperature dependence of G΄, G˝ and δ = tan-1(G˝/G΄) upon heating is shown in Figure

1, and reveal a glass transition at ~ 90 oC and an order-order transition TOOT at ~120 oC. The data

agree with low frequency measurements reported on a mixture of S-EB and SEBS, and show

6

similar behavior to that of the randomly oriented sample of SIS in reference 7. Because the

sample is randomly oriented, G΄ and G˝ of the cylindrical phase are higher than that of the

spherical phase. Hysteresis effects are observed on cooling (data not shown), as is usually

expected with these materials.

Atomic Force Microscopy. A Model 3100 AFM (Digital Instruments, Santa Barbara, CA)

attached to a Nanoscope IIIa controller with an electronic extender box at Boston University

Photonics Center was used for the present studies. The sample of SEBS 45% in mineral oil was

spin cast on a silicon wafer by diluting in toluene which evaporates during the spin casting

process. The spin cast sample was annealed at 110 oC for 24 hours. Just before AFM imaging

the sample was quick-frozen in liquid N2 to preserve the high temperature morphology and

imaged at ambient temperature using tapping mode. The height image from the AFM

measurement shown in Figure 2 reveals a well-ordered HEX morphology with a d-spacing of 35

nm between neighboring cylinders. The radius of the cylinder is estimated to be 10 nm. Note,

that due to tip broadening this is an overestimate of the actual cylinder radius.

7

Figure 2(a). Room temperature AFM height image (1μm x 1μm) from SEBS 45% in mineral

oil annealed at 110 oC and quick frozen to preserve the annealed morphology shows

hexagonally packed cylinders. The 3-dimensional rendering in (b) shows that the cylinders are

oriented perpendicular to the substrate. The contrast in the image is due to the differential

hardness of the glassy PS cores (bright) and the soft PEB matrix (dark). A section analysis

gives the height profile (c) along the line shown in the lower image (d) indicating that the

spacing between the cylinders is 35 nm. The power spectrum of the image (e) exhibits good

order along the line indicated in the image shown in (d).

Small Angle X-ray Scattering. Time-resolved SAXS experiments were conducted at

beamline X27C of the National Synchrotron Light Source (NSLS) of Brookhaven National

Laboratory, using X-ray of wavelength λ = 0.1366 nm (9.01 keV) with energy resolution dE/E =

8

1.1%. The scattering intensity was recorded on a 2 dimensional MAR CCD detector with an

array of 1024 x 1024. For the solution samples used here, the scattering patterns were isotropic,

so an azimuthal average was done to obtain the scattered intensity I(q) as a function of the

scattering wavenumber, q over the range 0.1 < q < 3 nm-1. The sample was loaded into a custom

designed cell made of a copper plate with a 0.6 cm-diameter hole covered with two thin flat

Kapton windows. A custom-designed computer controlled Peltier heater/cooler module

connected to the sample cell was used to change the temperature either rapidly (temperature

jump) or at a constant rate (temperature ramp). The desired temperature was reached within 1

minute with the Peltier module. Typically, the scattering intensity profile I(q) was recorded for

approximately 10 s per frame (includes data acquisition time and the time to read the array), and

the total time of each run was 1 to 2 hours. All scattering data were corrected by normalizing by

the incident beam intensity, and subtracting the scattering from the solvent. This procedure

allows us to compare the relative intensity from different frames following a temperature jump or

ramp, but it is important to note that the intensity data are not calibrated against a standard and

hence do not give the absolute intensity. It is also important to note that the cell used here allows

has flexible Kapton windows. Details of the experimental set up and data processing are

described in our previous work on disorder to BCC kinetics in SEBS triblock copolymer solution

in mineral oil33.

9

Results and Discussion

Part I: Small Angle X-ray Scattering Experiments

Figure 3 shows long-time averaged SAXS data from the 45% SEBS in mineral oil sample at

110 oC and 155 oC averaged for 20 min. These data clearly confirm that at 110 oC the sample is

in the HEX phase (peaks at relative positions of 7:4:3:1 ), while at 155 oC it is in the

BCC phase (peaks at relative positions of 7:6:5:4:3:2:1 ).

Figure 3. Long-time averaged SAXS data (in arbitrary units (a.u.)) from 45% SEBS in

mineral oil at 110 oC and 155 oC. The first 5 peaks for HEX at 110 oC and the first 7

peaks for BCC at 155 oC are marked. The plot is divided into two parts with an

enlarged intensity scale for the high-q region to clearly display the higher peaks.

Identification of HEX to BCC Transition by Temperature Ramp Measurement. To

determine the OOT temperature, SAXS data were acquired while the sample was being heated at

a constant rate of 1 oC/min from 70 oC to 180 oC, as shown in Figure 4.

10

Figure 4. Time evolution of the scattered intensity I(q) during heating of the SEBS

o o oC/min. sample from 70 C to 180 C at a constant rate of 1

Initially at 70 oC the sample is in a glassy state and a HEX phase can be clearly identified

around 90 oC. The relative positions of the peaks are characteristic of the HEX structure in the

temperature range of 90 - 120 oC and of the BCC structure at higher temperatures, with a clear

transition in the vicinity of 120 - 130 oC. The appearance of a peak ( 2 peak) at q2 = √2 q1,

where q1 denotes the primary peak position, is indicative of the transition from the HEX to BCC

phase. To identify the transition temperature we plot the peak intensity (I1 and I2) and position

(q1 and q2) of the first two Bragg peaks as a function of the temperature, as shown in Figure 5.

The peak intensity, position and width of the primary peak were determined by a fitting

procedure described in previous publications from our group34 . The values obtained by the

fitting procedure are in excellent agreement with parameters determined by direct inspection of

the data. The peak position is determined to an accuracy of 0.004 nm-1. The parameters for the

second peak were directly obtained from the data, because the peak is weak and that makes the

fitting method unreliable.

11

Figure 5. Temperature dependence of the first two Bragg peaks measured from the

temperature ramp data shown in Fig. 3. (a) Primary peak intensity I1, (b) position q1.

(c, d) The same parameters for the √2 peak. The discrete jumps in the peak position are

due to the fact that one pixel at the detector corresponds to 0.004 nm-1.

We observe that the intensity of the primary peak reaches a local minimum at ~127 oC, and

the 2 peak (characteristic of the BCC structure) first appears at ~127 oC. From this we identify

that the HEX BCC order-order transition temperature TOOT ~ 127 ± 2 oC. Above this

temperature I2 increases rapidly. We also found that the intensity, I3, of the √3 peak (not shown)

decreases above this temperature. These changes indicate the conversion of the HEX state to the

BCC state. We observe that q1 increases with temperature, implying that the lattice constant

decreases with increasing temperature. Similar shift in peak position with increasing temperature,

has also been noted in earlier experimental work 3, 4, , , , 7 9 19 35 and also been predicted by theory36.

12

The intensity I2 reaches a maximum at 150 oC, and as discussed later, we identify this

temperature as the spinodal temperature (Ts) corresponding to the metastability limit of HEX in

BCC. We also observed that at about 180 oC the intensity I2 decreases rapidly, and the peaks

become broader, indicating the onset of an order-disorder transition. It is important to know that

the temperature ramp method over-estimates the transition temperatures since the results depend

on the rate of heating.

Kinetics of the HEX BCC Transition. To study the kinetics of this order-order transition

we made time-resolved SAXS measurements following a temperature jump (T-jump) from a

sample annealed at a fixed initial temperature Ti = 110 oC in the HEX phase to various final

temperatures Tf above TOOT. The kinetics was measured for about 2 hours. It took about 60 -

160s for the system to respond to the T-jump and reach the desired final temperature. The

temperature equilibration time, t0 depends on the magnitude of the temperature jump, defined as

ΔT = Tf - Ti, as shown later in Table 1. Typical SAXS results for the early stages of the T-jump

with Tf = 135 oC are shown in Figure 6.

o o Figure 6. Time evolution of the SAXS intensity following a T-jump from 110 C to 135 C.

13

Analysis of SAXS Data. To follow the details of the time evolution we analyzed the primary

peak’s position (q), intensity (I), and width (w) as a function of time. Figure 7 shows the time

evolution of these parameters for the T-jump from 110 oC to 135 oC. Immediately following the

T-jump, the temperature of the sample changes rapidly and reaches the final temperature, Tf in

time t0 = 60 s (see Figure 7c). During this initial time the structure is that of HEX lattice and the

intensity drops very rapidly, in response to the rapid change in temperature. Further isothermal

time evolution exhibits three stages:

Stage I (t0 < t < t1), where the structure still shows peaks characteristic of the HEX lattice, but

the peak positions shift rapidly to higher values, indicating that the cylinders are moving closer

together. During this stage the primary peak intensity I1 (of HEX (100) peak) decreases, and the

slope of I1 versus t graph shows a sharp change at t1 = 120 s;

Stage II (t1 < t < t2), in this period the primary peak intensity first grows then decreases,

reaching a second local minimum at t2 = 1400 s;

Stage III (t > t2), during which the intensity of the primary peak increases monotonically,

reaching a stable value for the BCC phase eventually.

The separation into three stages is also supported by the non-monotonic behavior of the peak

width, w1 which has a minimum at t1 and maximum at t2. The peak position q1 increases rapidly

up to t1 and slowly thereafter, becoming more or less stable after t2. A similar approach was used

by Sota et al in the analysis of the transition of BCC to HEX for a shallow quench. They also

observed three stages after the temperature incubation time, with two steps corresponding to the

phase transition period and the last one to the growth of the HEX structure.

14

Figure 7. Time evolution of (a) the intensity (I

Further support for the onset of BCC structure is obtained from analyzing the 2 peak

corresponding to the (200) reflection of the BCC lattice which appears around t1 and is clearly

identifiable around t2 (as shown in Figure 8a). The intensity I2 increases after t1, while its width

narrows as shown in Figure 8b.

1), (b) the position (q1) and width (w1) of

the primary Bragg peak following a T-jump from 110 o oC to 135 C (see the SAXS data

shown in Figure 6). (c) The temperature equilibration during the very early stages of the

jump. Note the temperature data is only shown for the first 200 seconds, although the

temperature was recorded throughout the experiment. The times t , t and t0 1 2 (see text for

definition) indicating the different stages in the time evolution are labeled.

15

Figure 8. The time evolution of (a) the intensity (I ), and (b) the width (w2 2) and position

(q

The first stage corresponds to a pre-transitional incubation period where HEX cylinders

move closer together; the second represents the phase transition from HEX cylinders to BCC

spheres, i.e., the cylinders are modulated and modulation amplitude grows until cylinders break

up into the spheres in BCC symmetry; and the third represents the growth of the BCC sphere

phase. The presence of the incubation period indicates a nucleation and growth mechanism,

instead of spinodal decomposition.

Dependence of the Kinetics on the Depth of the T-jump. Figure 9 shows the data for the

peak parameters for the various T-jump measurements.

2) of the 2 peak following a T-jump from 110 oC to 135 oC. The solid line in (a) is the

fit to Eq. (1) describing a stretched exponential growth of the BCC phase, as described

later in the text. The position q2 is determined to an accuracy of ~ 0.004 nm-1.

16

Figure 9. Time evolution of peak intensity (I1), width (w1), and position (q1) of the primary

peak of SAXS data for various T-jumps (ΔT = 25, 30 and 35 oC as indicated). The initial

temperature was fixed at 110 oC for all T-jumps. All T-jumps show a 3-stage time evolution

following the initial temperature equilibration period.

The T-jumps with ΔT = 30 and 35 oC show similar behavior as described above for ΔT = 25

oC with three time regimes. In Fig.9, the intensity is normalized by the initial intensity at t = 0.

and the figure shows that the intensity at late time (around 5000 s) increases with increasing ΔT,

while the peak width narrows. The intensity depends on the amount of transformation material as

well as on the size of growing microdomains. In the limited duration of our experiment the

coarsening process is not complete. For the larger T-jump (with larger ΔT), the coarsening

17

process is faster and hence the domains are bigger in size, giving a higher intensity. The larger

size of domain for larger T-jump is also confirmed by the narrower peak width (shown in Fig.9b).

The peak position data (Fig.9c) is consistent with previous measurements3, , , , , 4 7 9 19 35 which show

that lattice constant decreases with increasing temperature in both HEX and BCC phase. The

time t1 before which the HEX cylinders move closer together is almost independent of

temperature; the time t2 decreases with increasing temperature jump (see Table 1).

Table 1. The temperature dependence of different time regimes and the position of the primary

peak for various T- jumps in the HEX to BCC transition.

ΔT (oC) to (s) t1 (s) t2 (s) q1(BCC)/q1(HEX) q1 (BCC) (nm-1)

25 60 120 1400 1.09 0.224

30 60 150 930 1.11 0.226

35 100 130 560 1.13 0.230

45 120 200 1.14 0.234

120 160 t΄ = 1500 s 1.20 0.255

This indicates that with increasing ΔT the transformation occurs faster, as is usually expected

due to the increased thermodynamic driving force for larger jump in temperature. The

temperature dependence of (t2 - t1), the transition period, for the ΔT = 25, 30 and 35 oC jumps is

shown in Figure 10 and extrapolated to zero by linear fitting. The transition period vanishes at

18

Figure 10. The dependence of t2 - t1, the transition period for HEX to BCC transformation

on the depth of the T-jump (ΔT). The straight line is a linear fit to the data which

extrapolates to zero at ΔT = 40 oC corresponding to the limit of metastability for the HEX

phase at 150 oC.

around ΔT = 40 oC corresponding to Tf = 150 oC, which indicates that the transition mechanism

is different above and below 150 oC. We estimate that this temperature is in the vicinity of the

limit of metastability of HEX in BCC, i.e. the spinodal temperature (Ts). At this temperature we

also observed that I2 is a maximum in the temperature ramp measurement (see Figure 5). To

make a semi-quantitative comparison with theory we examine the ratio of Ts /TOOT . Since we

could not find any calculated phase diagram for a triblock copolymer in a selective solvent we

make a qualitative comparison with predictions for diblocks. As pointed out by Matsen et al37

the melt of an ABA triblock exhibits similar phase diagram as the AB/2 diblock formed by

snipping the triblock in half, but differ in mechanical property due to the formation of bridges of

the middle B block between two outer A domains. The selectivity of mineral oil to middle block

PEB further enhances the microphase separation tendency between PS and PEB block, therefore

our triblock SEBS solution has similar behavior as the melt of S-EB, although χN for phase

19

boundaries will be at slightly different values than the diblock prediction. Matsen predicted a

narrow window for diblock copolymer melt in which the mechanism would be that of spinodal

decomposition. In his calculation, for a diblock copolymer melt of composition fraction f = 0.28,

the OOT of HEX to BCC is at (χN)OOT ≈ 16.4 and the spinodal is at (χN)s ≈ 15.3. We find that

the ratio of the Ts /TOOT obtained from the SAXS data for SEBS in mineral oil is 423K/400K ≈

1.06 which agrees quite well with Matsen’s prediction of (χN)OOT /(χN)s ≈ 1.07 for a diblock

melt with f = 0.28 (assuming that χ ~ 1/T, where T is the absolute temperature).

Growth of the BCC Structure Following a T-jump. The time evolution of I2 can be used

as a measure of the growth of the BCC phase. The peak intensity I2 increases with time

approaching a steady value finally. The time evolution of I2 could be fit by the stretched

exponential formula:

)1))(()(()()( )/)((00

0nttetItItItI τ−−

∞ −−=− . (1)

Figure 11 shows the normalized intensity of the √2 peak, defined as [I (t) – I (t0)] / [I (t∞) – I

(t0)], and the results of the fit to Eq. (1). The normalized data almost coincide for the jumps with

ΔT = 25, 30 oC but are identifiably different for ΔT = 35 and 45 oC. The fitting parameters τ and

n are listed in Table 2. The characteristic time τ decreases with increasing ΔT as expected. For a

T-jump above the spinodal line, ΔT = 45 oC, τ is much smaller, indicative of a faster process of

the transition due to the larger driving force for deeper jumps. The exponent n is close to 1 for

ΔT = 35 and 45 oC (corresponding to exponential behavior), and departs slightly from

exponential for the shallower jumps (1.2 -1.3).

20

Figure 11. The time evolution of the normalized intensity of the √2 peak data for various T-

jumps as indicated by the different symbols. The results of a stretched exponential fit (Eq.

(1)) to the data are shown with the solid lines. The normalized data for ΔT = 25, 30 oC are

very close together indicating a very weak temperature dependence in this temperature

range.

Table 2. The parameters for the stretched exponential fit to Eq. (1) of √2 peak intensity I2 for

different T-jumps.

ΔT (oC) τ (s) n

25 1918 1.3

30 1911 1.2

35 1185 0.9

45 147 0.9

21

The overall growth curves are similar to the predictions of the TDGL calculations for the

HEX to BCC transformation via the growth of anisotropic fluctuations. The interpretation of the

SAXS data in terms of the growth of anisotropic fluctuations will be discussed in the second part

of this work. The stretched exponential fit to the data for the shallow jumps which are below the

metastability limit is consistent with a nucleation and growth mechanism, usually described by

the Avrami equation38 which has the same form as Eq. (1). It is important to note that in this

transition ripples along the cylinder nucleate and form spherical micelles, as has been described

in detail by Matsen.

Kinetics of a T-jump Above the Spinodal. For a deep jump with ΔT = 45 oC (i.e. to a

temperature above Ts), we observed a qualitatively different behavior than for the shallow jumps

below the spinodal, as shown in Figure 12.

Figure 12. Time evolution of the intensity I, width w and position q of the primary peak and

the √2 peak of SAXS data for T-jump from 110 oC to 155 oC.

22

For the ΔT = 45 oC jump the two times t1 and t2 cannot be distinguished and no intermediate

stage could be detected. The transformation from the HEX cylinders to BCC spheres occurs via

the mechanism called Model C in the Hohenberg-Halperin 39 classification scheme which

involves both a non-conserved field (due to the symmetry changing transition) and a conserved

field (composition of block copolymer as well as concentration are both conserved). Although

the conservation condition is similar to the spinodal decomposition in a binary polymer blend40,

the symmetry breaking is unique to the block copolymer. The symmetry-breaking feature is

similar to that observed in continuous ordering in metallic alloys, but there is no conservation

condition in that case 41 . Unlike the Cahn-Hilliard equation 42 used for describing spinodal

decomposition in a polymer blend, there is no simple analytical expression for predicting the

time evolution of the scattering function for Model C. As expected the transition for the deeper

jump with ΔT = 45 oC occurs faster than for the shallower jumps with ΔT < 40 oC due to the

larger thermodynamic driving force.

Jump from HEX to Disorder State Exhibits a Transient BCC Phase. We also made a

very deep T-jump measurement from the HEX phase at 110 oC to 230 oC at which the sample is

eventually disordered. For this very deep jump with ΔT = 120 oC as shown in Figure 13, the peak

intensity I1 decreases rapidly during the time t0 in which the sample temperature equilibrates.

After this time a BCC phase can be identified, which persists for about 1500 seconds. In this

time interval the 2 peak appears and grows in intensity and gets slightly narrower, while the

primary peak changes little in intensity or width.

23

Figure 13. Time-evolution of the intensity and width of the primary peak and 2 peak

for a very deep T-jump from 110 oC to 230 oC. The HEX structure transforms initially to a

BCC state which persists for about 1500 seconds. The disordering time t΄ = 1500 s, after

which the system transforms from BCC to a disordered micellar sphere state, is indicated.

After 1500 seconds the sample becomes disordered as evidenced by the decreasing intensity and

broadening of both peaks. Hence, we conclude that the HEX cylinders first undergo an order-

order transition forming a transient BCC sphere phase and then order-disorder transition occurs

at around 1500 seconds identified as t′ in Figure 13. Interestingly, in the reverse transition from

disordered spheres to HEX cylinders reported by Sota et al 43 a transient BCC sphere state was

seen for a shallow quench below TOOT. A transient BCC state was also observed in the disorder

to FCC transition in a SI block copolymer in tetradecane, an isoprene selective solvent.

24

Geometrical Characteristics of the HEX Cylinder and BCC Sphere Phases. From Table

1 we note that the ratio q1(BCC)/q1(HEX) = d100/ d110 (where d100 and d110 denote the principal

lattice spacing of the HEX and BCC phases = 2π/q1) increases with increasing ΔT. For large ΔT,

the ratio is bigger than the theoretical prediction of 1.08 for melts, and is also larger than the

values reported from experiments in melts 4, 7. From the positions of the primary Bragg peak we

can estimate the principal lattice spacing d* = 2π/q1, and find that it varies from 30.9 nm for the

HEX cylinder structure to 25.3 - 28.0 nm for the sphere BCC structure. The position of the first

minimum (qmin) of the form factor can be related to the core radius via r = 4.49/qmin for sphere

and r = 3.83/qmin for cylinder. We determined this minimum by examining the SAXS data on a

greater magnification than shown in Figure 2. The radius of the cylindrical domain estimated

using this relationship is about 8 nm (for T = 110 oC), while the radius of sphere is about 10 nm

(for T = 135 oC). We note that the cylinder radius value obtained from the AFM image of the

HEX phase is a little larger than that calculated from the SAXS data as expected due to tip

broadening effects. The position qmin in the SAXS data shifts continuously to lower value as time

increases reflecting the increase in the radius from cylinder to sphere. The average end-to-end

length of a PS chain of the SEBS triblock treated as Gaussian is LPS = aPS NPS0.5 = 8 nm using aPS

= 0.71 nm and NPS = 134 as the number of PS monomers in each PS chain44. This length is very

close to the radius of the cylinder but smaller than the radius of the sphere, suggesting that in the

sphere phase there may be some solvent in the core and/or the chain may be stretched. This is

plausible because solvent selectivity decreases with increasing temperature.

Part II: Model Calculation of Scattering Intensity

25

In this section we discuss a geometrical model to calculate the scattering from the rippled

cylinders that form in the process of the HEX to BCC transition. This model is applicable to

explaining the temperature jumps beyond the spinodal, where the cylinders are unstable with

respect to modulation, and thus ripples form over the entire length of the cylinders which are

correlated with their neighbors. Figure 14 schematically shows the geometrical model for the

transition from HEX cylinders to BCC spheres (adapted from Laradji et al ) with seven

unmodulated cylinders in 2-d HEX lattice as initial state.

Figure 14. Schematic illustration of the transition from HEX to BCC. (a) The initial state

of seven unmodulated HEX cylinders. (b) At the intermediate stage, the cylinders are

modulated in a coupled way as discussed in the text. As the amplitude of the modulation A

grows the rippled cylinders break up into spheres as shown in (c). When the distance

between neighboring cylinders approaches 3/22 λ=d , a commensurate BCC structure is

formed. A BCC cube is shown as a visual aid.

26

To represent the ripple due to the anisotropic fluctuations we modulate the radius of a

cylinder oriented along the z direction by a transverse wave along the z axis as,

r(z) = r0 + A cos (2π z / λ + φ). (2)

Here r is the radius of cylinder, A and λ are modulation amplitude and wavelength respectively,

and φ is the phase of the modulation. The epitaxial relation for the HEX to BCC transition

requires that the <001> direction corresponding to the axis of the cylinder becomes the <111>

direction of the BCC lattice, and the three (100) planes of HEX transform into the (110) planes

of BCC structure. In order to obtain the epitaxial relationship the modulation of cylinders has to

be coupled as shown in Figure 14b.

The question arises as to how to select the phase shifts φ between neighboring cylinders.

Intuitively it is obvious that if the bulges of two neighboring cylinder are in phase then there will

be unfavorable steric interactions. We formalize this idea by a simple calculation of minimizing

the overlap volume between neighboring modulated cylinders. Obviously the configuration with

minimum overlap volume is the most favorable. The overlap volume shown in Figure 15 was

calculated by considering three adjacent cylinders on an equilateral triangular lattice. If we set

one of the 3 cylinders as having phase shift 0, then the other two have phase shifts φ and 2φ

respectively. This implies a constant difference between neighboring cylinders counted in a

cyclic manner. The total overlap volume for the system is N times of that obtained for this

triangular unit, where N is the total number of HEX cylinders. To calculate this volume we

assume that the cylinders can be sectioned as hard-discs with a radius varying in z direction as

given by Eq. (2). In the block copolymer SEBS in mineral oil, the hard-disk radius rhs consists of

two parts: r0 which corresponds to the radius of the PS cylindrical core, and rhs - r0 which

27

represents the swollen corona formed by the loops and bridges of the PEB chains. In this sense

the hard disk represents the exclude volume interaction. By integrating over the length of the

cylinder we obtain the overlap volume as a function of φ.

Figure 15. Overlap volume of 3 adjacent modulated cylinders in an equilateral triangle

lattice with phase shifts 0, φ, and 2φ respectively. The model parameters are set as: rhs =

15.5 nm, λ = 33 nm, A = 6.5 nm, d = 31.1 nm and length of cylinder L = 1000 nm. The

minima occur at φ = 2π/3 and 4π/3, which are equivalent to each other.

The results obtained by numerical integration, shown in Figure 15, clearly indicate minima at φ

= 2π/3 and 4π/3. We note that these two values are equivalent45. If we alternate the phase shift φ

of the 6 surrounding cylinders as 4π/3 and 8π/3 with respect to the center cylinder (φ = 0), then

the resulting spheres will arrange on a BCC lattice (Figure 14b), and the epitaxy is automatically

satisfied. As the amplitude of the ripple grows and reaches the maximum, A = r0, modulated

cylinders break into spheroidal “pearls” as shown in Figure 14c. When the spacing between

neighboring cylinders, 3/22 λ=d , a commensurate BCC structure is formed with lattice

constant d5.13/2 == λ . To keep the volume of rippled cylinder conserved as the amplitude

of the modulation, A, grows, r0 decreases as

28

[ ] 2/)/2/()/2sin(1)0()( 2200 λπλπ LLAArAr −−== , (3)

where L is the length of rippled cylinder. We note that an FCC phase will form if we use

6/2λ=d and the same phase shifts as above. The yield of the twinned BCC transformed from

HEX has been predicted theoretically46 and demonstrated experimentally6,7. According to our

model, the twinned BCC arises from two sets of rippled cylinders with clockwise and counter-

clockwise phase shifts of (0, 4π/3, 8π/3) which are mirror to each other along a HEX (100) or

(110) plane.

Scattering Intensity Calculation. The calculation of the scattering function of HEX

modulated cylinders proceeds in three steps. First we calculate the form factor p( ) of a single

rippled cylinder with a definite orientation relative to the scattering wavevector

qr

)2

sin(4 θλπ

=qr

where θ is the scattering angle. Next we calculate the scattering intensity from an oriented

domain using the form factor of rippled cylinder and the structure factor of the HEX lattice, and

finally obtain the azimuthally averaged scattering intensity by averaging over all orientations.

The form factor of a single rippled cylinder oriented along the z-axis using cylindrical

coordinates, is obtained by integrating over the volume Vcyl of the rippled cylinder

rdrqiqpcylV

rrrr 3)exp()( ∫ ⋅−= . (4)

The integration in Eq. (4) over the circular polar coordinates can be performed analytically,

giving

dzzArqBq

zAriqzqpL

L]sin))/2sin(([

sin))/2sin((2)cosexp()( 01

02/

2/αλπ

αλππ

α ++

−= ∫−

r, (5)

29

where BB1(x) is the first order Bessel function, α is the angle between qr and the z-axis (the polar

angle in cylindrical coordinates), and q is the magnitude of qr .

The structure factor S( ) of a HEX lattice of N cylinders all oriented at an angle α relative to qqr r

is given by:

∑−

=

Δ⋅−+=1

1)exp(1)(

N

iirqiqS rrr

, (6)

where denotes the position vector of the i-th cylinder 0rrr iirrr

−=Δ irr

relative to a chosen cylinder

denoted as in the HEX array. For one crystalline domain all the cylinders make the same

angle α with q so the contribution of a single domain with N cylinders to the scattering is |S(

0rr

r qr ).

p( )|qr 2 . Since the experimental data described earlier is for unoriented samples and we observed

a uniform azimuthal distribution of the scattered intensity we assume that the crystalline domains

are randomly oriented. Hence, the azimuthally averaged scattered intensity I(q) is calculated by

numerical integration over the angular space as:

∫ ⋅⋅=π

αα0

2 sin|)()(|21)( dqpqSqI rr

. (7)

The parameters for the size and spacing were obtained from the experimental data. The

radius of the cylinder r0 was taken as 8 nm, and that of the sphere as 10 nm. The spacing

between neighboring cylinders d was determined from the peak position using Eq. (8). Since it is

not possible to determine the length of the cylinder from the SAXS data, we chose L = 1000 nm.

This value is of the same order as usually seen in TEM images of block copolymers. The number

of cylinders N in one domain was chosen to get the width of I(q) in reasonable agreement with

experiment. The larger the number of cylinders in one domain the narrower is the calculated

30

peak. We found reasonable agreement for widths with N = 381. A typical calculation based on

the model described above is shown in Figure 16 with A = 4.5 nm, λ = 32.9 nm, d = 32.3 nm.

Figure 16. A typical scattering intensity calculated from the model. The parameters of the

calculation are N = 381, L = 1000nm, r0 = 8nm, λ = 32.9nm, A = 4.5nm and d = 32.3nm.

The main peak (q ) and the side-peak (indicated as q1 1*) both contribute to the scattering

intensity of the primary peak. The 32 peak and peak are also indicated.

The results are qualitatively unchanged on varying L, λ, d and N, although the peak positions and

intensities changed with d, A and λ. The primary peak, the 2 peak and 3 peak are clearly

displayed in Figure 16.

We note that the primary peak is split into two peaks, a main peak (q1) and a side-peak (q1*).

The main peak arises from HEX (100) plane with peak position

( )dq 341 π= . (8)

31

The three HEX (100) planes will become the three BCC (110) planes parallel to the cylinder axis.

The side peak arises from the other three BCC (110) planes that are not parallel to cylinder axis

with peak position

22*

11

942

λπ +=

dq . (9)

According to our model, the three BCC (100) planes parallel to the cylinder axis will not be

identical to the three non-parallel BCC (110) planes unless the condition 3/22 λ=d is

satisfied and modulated cylinders fully break up into BCC spheres. In other words, the main and

side peak will not coincide until d equals 3/22 λ . Furthermore, the intensity of the side peak

q1* is weaker than that of main peak q1 and it grows as the modulation amplitude A increases.

Eventually when the amplitude A approaches the maximum value r0, i.e., when the modulated

cylinders fully break up into spheres, the intensity of the side peak equals that of the main peak.

This phenomenon of two principal peaks due to the mismatch of d and λ has been addressed by

Matsen and can be observed in the experimental data on SIS melt reported by Ryu and Lodge.

The side peak is clearly seen in Fig. 7 of reference 7. From the positions of the two peaks in that

figure, we can calculate d and λ for the SIS melt using Eqs. (8) and (9). We obtain nm35≈λ

which exactly agrees with their TEM observation. In fact, the side peak q1* corresponds to the

two sets of 6 fluctuation spots (total 12 spots for the twinned BCC) in a reciprocal space sphere

developed by Qi and Wang, and the main peak q1 corresponds to 6 spots of original HEX

principal peak. However, the BCC structure would produce the same scattering pattern but all

spots would be equally intense because they come from identical reflections. In contrast, for

modulated cylinders the peak position is mismatched and the intensity of fluctuation spots is

weaker than that of the original HEX principal peak. Fig. 14 of reference 7 also shows the

32

appearance of 4 new weak spots in the x-z plane in q space, which grow in intensity with

annealing time, providing a clear signature of the formation and growth of modulated cylinders.

The behavior of primary peak width of SAXS data with T-jump ΔT = 45 oC shown in

Fig.12b is an indication of the 2-peak splitting. The emergence of the second peak q1* (related to

d and λ) broadens the primary peak because the position of the second peak does not coincide

exactly with the first peak due to the mismatch of d and λ. As the modulation amplitude A

increases, the second peak grows, meanwhile, the mismatch of d and λ decreases as d approaches

to √2λ/1.5. Thus, as A increases, the primary peak first becomes broader and then narrows later,

which is consistent with the experimental result shown in Fig.12b.

Due to the resolution limit of our experiment, we were not able to resolve the two-peak

splitting visually from the SAXS data. However, the peak profile was asymmetric as shown in

Figure 17a. To confirm that this asymmetry is due to a second peak close to the primary peak, we

used a simple procedure of reflecting the data below the maximum position and then subtracting

the reflected curve from the original data. As seen from Figure 17a, the subtracted intensity

indicates a second peak whose intensity grows with time. Due to the uncertainty of the choice of

the reflection position, this method is not suitable to determine the position of the two peaks

quantitatively. To obtain the peak positions we fit the primary peak of the experimental data with

2 Gaussian peaks (shown in Figure 17b). The time-evolution of d and λ (shown in Figure 17c)

was determined from the positions of the two peaks using Eqs. (8) and (9). Figure 17b shows that

the two initially indistinguishable peaks (within the experimental resolution) separate with

increasing time and then eventually merge together. Note that at t = t2 = 200 s, when BCC is

33

formed, the two peaks are not merged indicating that d and λ do not satisfy the commensuration

Figure 17. (a) The primary peak profile of a few frames of SAXS data for T-jump ΔT = 45 oC

(empty symbols and thick lines). Also shown are the mirror reflections (filled symbols) and

the subtractions (empty symbols and thin lines). Frames at different times are shifted for

clarity. (b) The time evolution of the positions of the two peaks obtained by fitting the

primary peak with 2 Gaussians. (c) λ and d obtained from the peak positions using Eqs. (8)

and (9). To illustrate the effect of commensuration d is multiplied by 1.5/√2.

34

relationship at this time. The spheres continue to move eventually forming a well-defined BCC

structure with d = 31 nm at around 1500 s, which agrees with the prediction by Matsen. We note

that in calculating d and λ the peak with higher q value is identified as the side peak q1* because

the corresponding intensity is lower than of the other one and grows as time increases.

Comparison of Observed Kinetics with the Model Calculation. As we addressed before,

our model is best suited to explaining the temperature jump beyond the spinodal (ΔT = 45 oC),

where all the cylinders ripple simultaneously. The situation is more complicated for the

nucleation and growth scenario (shallow temperature jump below the spinodal) because some

parts of cylinders would develop ripples while others would remain unmodulated as discussed by

Matsen and the front would advance with time.

In order to compare the model to the experiment with T-jump ΔT = 45 oC we use the values of d

and λ obtained from the two Gaussian peaks fitting procedure (see Figure 17 and Table 3).

Table 3. The parameters of the rippled cylinders (A, d, and λ) used for model scattering intensity

calculation.

A (nm) 0 1 2.5 3.5 4 4.5 5 5.8 6.5 6.5

d (nm) 35.6 34.3 33.8 33.6 33.3 33.1 32.8 32.6 32.3 31

λ (nm) -- 35.2 34.5 34 33.5 32.9 32.4 32 31.8 32.8

There is no direct way to obtain the amplitude as a function of time from the data. As a

simple approach the value of A is set such that during the transition period (t0 < t < t1 = t2), A

35

increases roughly linearly from 1 nm to 6.5 nm (r0 decreases to 6.5 nm as A = 6.5 nm). The

results of the numerical calculation of the scattering intensity are shown in Figure 18a along with

SAXS data for the T-jump ΔT = 45oC.

Figure 18. (a) The calculated scattering intensity at different values of the modulation

amplitude A as indicated. The parameters of calculation are N = 381, r0 = 8 nm, L = 1000

nm. The values of λ and d vary as A varies from 0 to 6.5 nm (see Table 3). (b) Selected

frames from t = 0 to 1500 s as indicated, from the time-dependent SAXS data for T-jump

ΔT = 45oC are shown for comparison with the calculation. The scattering curves are shifted

vertically for clarity.

The splitting of the primary peak of the calculation is clearly seen for the calculations with A

= 4 - 6.5 nm. The positions of the two peaks q1 and q1*, as well as their integrated intensity

36

(defined as the product of peak intensity and width) I 1 and I 1* obtained by a Gaussian fitting

procedure are plotted in Figure 19. The integrated intensity of the side peak I 1* increases with

increasing A, whereas I 1 decreases. When A = r0 = 6.5 nm, the intensities are equal to each other.

The dependence of the intensity on the amplitude agrees with the self-consistent field calculation

of Matsen (see Fig. 4 and 7 of ref. 23).

* *, and (b) integrated intensities I

Overall, the numerical calculation of scattering intensity of the model agrees well with the

SAXS data for T-jump ΔT = 45oC (shown in Figure 18b). We have not attempted a fit of the data

to the model calculation because of the uncertainties of determining L and the time dependence

Figure 19. (a) Peak positions q1 and q 1 and I 11 obtained

from 2-Gaussian fit of the primary peak for the model calculation (Figure 18a) as a

function of the modulation amplitude A. For comparison q * *, and I , I , q 1 11 1 obtained

from the SAXS data for the T-jump with ΔT = 45 oC at early times are shown in (c) and

(d). The solid lines are shown for guide to eye.

37

of the amplitude. From the calculation (Figure 18a), we can observe a very clear appearance of

√2 peak even at small amplitude A. Experimentally, the onset of √2 peak could signal the

formation of modulated cylinders with ripples at BCC symmetry and does not necessarily

indicate spheres in BCC phase.

Conclusions

We have examined the kinetics of HEX to BCC transition in the triblock copolymer SEBS

45% in mineral oil, a selective solvent for the middle PEB block, using time-resolved SAXS

measurements. Temperature-ramp SAXS data show that the HEX to BCC transition occurs at

~127 oC with the spinodal Ts at ~150 oC, and ODT at ~180 oC. By examining various T-jumps

with the sample initially at 110 oC, we were able to observe the nucleation and growth

mechanism driven kinetics for shallow T-jump and the spinodal decomposition with continuous

ordering for deep T-jump. Temperature-jump experiments starting from 110 oC show that the

nucleation and growth kinetics involves three stages. In the first stage, t0 < t < t1, the cylinders

get close to each other while remaining in HEX structure; the second stage, t1 < t < t2, is the

transition period, where the cylinders are modulated along their axis and eventually break into

spheres on a BCC lattice, and finally in the third stage, t > t2, the domains coalesce and the

fraction of material in the BCC state grows.. The transition time, t2 - t1, decreases linearly with

increasing ΔT and extrapolates to zero at ΔT = 40 oC, corresponding to a spinodal at 150 oC for

the metastability limit of HEX in BCC. For a deep T-jump to a temperature above the spinodal,

the first two stages merge together, and the HEX to BCC transition occurs via a mechanism

involving continuous ordering and spinodal decomposition. In this case, after the initial

38

temperature equilibration time t0, the HEX cylinders transform to BCC spheres until t = t2 (= t1)

and after that the BCC domains coalesce and grow. We also examined the kinetics of the HEX to

disordered spheres transition and observed that the system first transforms from HEX to BCC,

followed by the order-disorder transition.

To calculate the scattering during the transformation stage we have developed a geometrical

model based on the previous theoretical models of anisotropic fluctuations, according to which

the cylinders develop a transverse wave-like instability that grows with time leading to the

formation of spheres. We found that when the phase shift φ of 3 adjacent cylinders in the unit

cell are (0, 4π/3, 8π/3) the overlap volume is minimized and the centers of the spheroidal bulges

lie on a BCC lattice. This model automatically preserves the epitaxial relationship observed in

experiments, i.e., the cylinder axis becomes the <111> direction of BCC, and the (100) planes of

HEX transform into the (110) planes of BCC. The calculated scattering intensity of the model

agrees well with the time-resolved SAXS data for the T-jump above the spinodal. We found that

initially the wavelength λ of the modulation is incommensurate with the cylinder spacing d. This

leads to a splitting of the primary peak into two peaks which merge together when λ

= 2/5.1*d . The integrated intensity of the higher-q component increases while that of the

other one decreases as the modulation amplitude increases; becoming equal to each other as the

modulation approaches its maximum. Although the two peaks could not be directly resolved in

the SAXS data reported here, their presence was inferred from the asymmetric shape of the

primary peak. The calculations reported here further support the theoretical predictions from

previous studies concerning the mechanism of the HEX to BCC transition.

39

Acknowledgements

This research was supported by NSF Division of Materials Research Grant No. 0405628 to

R.B. The SAXS measurements were carried out at Beamlines X27C and X10A of NSLS,

Brookhaven National Laboratory which is supported by the U.S. Department of Energy, Division

of Materials Sciences and Division of Chemical Sciences, under Contract No. DE-AC02-

98CH10886. We thank Dr. Igor Sics and Dr. Lixia Rong of beamline X27C, and Steve Bennett

of beamline X10A for technical support at NSLS, and Randy Ewoldt of MIT for help with the

rheology measurements. We acknowledge the support of Boston University’s Scientific

Computation and Visualization group which is supported by NSF for computational resources.

We thank Professors Bill Klein and Karl Ludwig for many stimulating discussions.

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43

Table of Content (TOC) Graphic

Kinetics of HEX-BCC Transition in a Triblock Copolymer in a Selective Solvent: Time Resolved Small Angle X-ray Scattering Measurements and Model Calculations Minghai Li, Yongsheng Liu, Huifen Nie, Rama Bansil, Milos Steinhart

44